Literally, a well-oiled machine is one that has had oil put on its moving parts, so nothing sticks or scrapes and everything turns smoothly. That picture is where the useful meaning comes from.

Most of the time you meet this word, no oil is involved at all. We use it for an organisation, a team, a system, or a process that works beautifully — every part doing its job, nothing wasted, no waiting around, no arguments. A restaurant kitchen at dinner time where every cook knows exactly what to do; an election campaign where the volunteers, the money and the messages all arrive on time; a school office where your form is stamped and returned in five minutes. The praise is not about one clever person — it is about the whole thing fitting together and running without friction.

By far the most common way you will see it is in the fixed phrase 'like a well-oiled machine'. Honestly, that phrase is where this word lives; outside it, 'a well-oiled operation' and 'a well-oiled team' also sound natural, but many other combinations start to sound odd. It is warm, everyday English — fine in a work email, fine in the newspaper, fine when talking.

One more thing, so you are not caught out. In British English there is an old slang use where 'well-oiled' means drunk — he came home well-oiled after the office party. It is dated and joking, and it is not what you will usually meet, but it exists, so 'a well-oiled guest' is not a compliment.